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Série Scientifique

Scientific Series

2011s-30

Bargaining with Linked Disagreement Points

Justin Leroux, Walid Marrouch

CIRANO

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Partenaire financier

Bargaining with Linked Disagreement Points *

Justin Leroux†, Walid Marrouch

‡

Résumé / Abstract

Dans un contexte de négociations bilatérales, nous adressons la question de l’inter-connexion des

sujets de négociation dans un modèle coopératif à deux-sujets-deux-agents. Les axiomes que nous

proposons insistent sur le rôle des points de menace. Une famille de solutions ressort de l’analyse : les

solutions monotones à taux nets identiques. Chacune de ces solutions préconise une issue Pareto

efficace de sorte que les gains relatifs de chaque agent sont les mêmes pour les deux sujets de

négociation. De plus, ces règles récompensent les agents pour des améliorations de leurs pouvoirs de

négociation. Nous discutons nos résultats à la lumière des négociations de commerce international et

environnementales, qui sont souvent amenées à la table de négociations de manière liée.

Mots clés : Négociations Multi-sujets, inter-connexion des sujets, solutions

axiomatiques, point de Menace.

In the context of bilateral bargaining, we deal with issue linkage by developing a two-issue-two-player

cooperative bargaining model. In contrast to the traditional Nash bargaining literature, the axioms we

propose focus on the role of changes in the disagreement points. We characterize a new solution that

we call the Linked Disagreement Points (LDP) solution, which explicitly links the players’bargaining

powers on each issue. We then weaken our axioms in turn, and a family of bargaining rule stands out:

the monotonic equal net ratio solutions. These solutions point to Pareto efficient outcomes such that

the relative gains for players are equal across issues and reward the players for improving their

bargaining power over each issue. We discuss our results in light of international trade and

environmental negotiations, which are often put on the bargaining table in a linked fashion.

Keywords: Multi-issue bargaining, issue linkages, axiomatic solutions,

disagreement point.

Codes JEL: C78, Q56.

* We are grateful for insightful comments from Walter Bossert, Bernard Sinclair-Desgagné, and participants at

the Montreal Resource and Environmental Economics Workshop. This research was made possible in part thanks

to funding from SSHRC. † CIRANO and HEC Montréal, 3000 chemin de la Côte-Sainte-Catherine, Montréal, QC H3T 2A7 Montréal,

Canada. E-mail: justin.leroux@hec.ca. Phone: (+1) 514-340-6864. Fax: (+1) 514-340-6469. ‡ CIRANO Lebanese American University, School of Business, Department of Economics and Finance, Beirut,

Lebanon, P.O. Box 13-5053 / F-15. New York Office 475 Riverside Drive, Suite 1846 New-York, NY 10115-

0065-USA, Email: walid.marrouch@lau.edu.lb. Phone: (+961) 01 78 64 56 Ext:1517. Fax: (+961) 01 86 70 98.

1. INTRODUCTION

Many bargaining situations involve multiple issues at once. For instance, inter-

national trade and environmental negotiations have often been put on the bargain-

ing table in a linked fashion. From Montreal in 1987, through Kyoto (1997) and

Cartagena (2003), to Copenhagen in 2009 international environmental agreements

were negotiated with the lurking spectre of trade (dis)agreements like the WTO.

Multiple-issue bargaining is a complex process where negotiations often break

down and a non-cooperative outcome prevails. The non-cooperative outcomes re-

sulting from disagreement are typically Pareto-inferior. Keeping with our moti-

vating example from international trade, when trade negotiations break down theresult is a tari¤ war leading to substantial welfare losses.

In order to propose Pareto-improving recommendations, we model issue link-

ages using a cooperative bargaining model with two players and two issues. We

consider the linkage between issues by expliciting the relationships between dis-

agreement points and possible Pareto-improving outcomes where the �nal Pareto

e¢ cient outcome is known. As such, we consider bargaining situations where play-

ers are negotiating on the sharing of a surplus that is pre-de�ned at the outset of

the negotiation round(s). Stylized facts suggest that countries�negotiation powers

over each speci�c issue (trade or environment) play an important role in shaping

the overall outcome of international negotiations because they act as threat points.

For instance, trade wars and trade negotiations in the pre-NAFTA context were

driven by the parties�disagreement points. In this context, Glenn Harrison and

Elisabet Rutström (1991) compute the non-cooperative Nash equilibrium of the

trade protection game between the US and Europe and evaluate welfare relative

to it. The Nash equilibrium can then serve as a "natural measure of nation�s bar-

gaining strength when entering into international trade negotiations, [where] this

bargaining strength is based on relative gains and losses in a credible disagreement

outcome, which [they] interpret as the disagreement outcome" (p. 421). This bar-

gaining mechanism was also observed within the genetically modi�ed organisms

dispute in the years 2003-2006, which pitted the USA, Canada and Argentina on

one side and the European Union on the other and was settled in favor of the former

group, where negotiation power over trade favored the winners1 .

The traditional single-issue cooperative bargaining framework describes a feasi-

ble bargaining set X and a corresponding disagreement point dX . We consider the

family of bargaining problems where the feasible set is pre-determined prior to the

negotiation round(s) where the role played by the disagreement point is paramount.

Therefore, in contrast to the Nash bargaining framework, we normalize the set X

1For further information consult the WTO�s dispute database.http://www.wto.org/english/tratop_e/dispu_e/dispu_subjects_index_e.htm

1

to be a bargaining �cake� of size one but allow the disagreement points to vary.

(Figure 1) And, when considering simultaneous bargaining over two issues, X and

Y , we link the two bargaining problems. This linkage is achieved by considering

the relative bargaining power of the players over each issue and how they relate to

the other.

[FIGURE 1 HERE]

We propose speci�c axioms dealing with issue linkages in the presence of non-

normalized disagreement points and normalized bargaining sets. A �rst axiom,

Uniformity, formalizes the linkage between issues by asking that if the agents�

relative bargaining powers are identical across issues, this relationship alone should

drive the �nal outcome. As such it speci�es to what extent the issues can be treated

independently of the other. The second axiom, Invariance, is stronger and states

that total payo¤s should not be altered by a reallocation of each agent�s bargaining

power across issues. In other words, by focusing on the combined bargaining powers

of the agents�it speci�es to what extent the two issues can be treated as a single

one.

These two axioms together characterize a family of solutions that are variants

to what we call the Linked Disagreement Points (LDP) solution (Theorem 1).

Graphically, the LDP solution links the disagreement points of each issues in the

mirrored utility space and selects the intersection of that line with the Pareto

frontiers of each issue�s bargaining set.(Figure 2)

[FIGURE 2 HERE]

Next, we drop one axiom at a time and explore the type of solutions the other

allows. The Invariance axiom appears as the more prescriptive of the two, as it alone

leads to a representation result (Theorem 2). By contrast, the Uniformity axiom

requires additional properties to arrive at a tractable class of solutions. Therefore,

combining Uniformity with other mild axioms leads to a family of solutions, which

we call monotonic equal net ratio solutions. These solutions point to Pareto e¢ cient

outcomes such that the relative gains for players are equal across issues. Moreover,

they are monotonic in the sense that they reward the players for improving their

bargaining power over each issue (Theorem 3). The LDP solution belongs to this

family of solutions, but not the Nash bargaining solution as it violates Uniformity

and thus fails to satisfactorily convey issue linkage.

2

The paper unfolds as follows. Section 2 discusses our theoretical contribution

in light of the related literature. Section 3 presents the two-issue bilateral bar-

gaining model and the �rst characterization results. In Section 4, we explore the

consequences of weakening the two main axioms in turn. Section 5 concludes.

2. RELATED LITERATURE

The literature on bargaining is made up of two strands: one follows a non-

cooperative approach à la Rubinstein (1982) and the other follows a cooperative

or axiomatic approach à la Nash (1950). Our work belongs to the latter. This

literature includes a number of attempts at modeling multiple-issue bargaining,

which have mostly ignored the importance of changes in the disagreement points

and the resulting spillovers between issues. An important general observation about

multiple-issue bargaining is that cooperative models have remained silent about

the role played by disagreement points, which are typically normalized to zero.

Meanwhile, stylized facts discussed earlier suggest that these points are pivotal in

negotiations since they acts as threat points2 . Moreover, the idea of concessions

exchange that arises in non-cooperative models is also neglected in those cooperative

models.

Most of the theoretical work on multiple-issue bargaining uses two-player models

and generalizes existing solutions from single-issue bargaining. This is done by

proposing new axioms that generalize or replace the classical ones found in the

literature on single-issue bargaining. When agents�preferences are represented by

a utility function it is assumed that utilities are additive across issues. Clara Ponsatí

and Joel Watson (1997) generalize the Nash solution and the symmetric utilitarian

solution. Michael Peters (1986), generalizes Ehud Kalai�s (1977) extended family

of proportional solutions and John Harsanyi and Reihard Selten�s (1972) extended

family of non-symmetric Nash solutions. Another approach has been more recently

proposed by Mármol and Ponsatí (2008) and uses maximin and leximin preferences

when information about preference is limited or when those preferences do not

admit a utility representation. This work follows Walter Bossert et al. (1996)

and Bossert and Peters (2001) by modeling the global bargaining problem as the

Cartesian product of classical (single issue) bargaining problems.

Cooperative bargaining problems invite three possible families of axioms. First,

there are axioms that are related to changes in the bargaining set, where the fo-

cus is on bargaining situation under variable bargaining trophies. These appear in

Peters (1985, 1986), and Ponsatí and Watson (1997) among others, where disagree-

ment points are normalized to zero. Second, there are axioms related to changes

in the population on which the literature has been mostly silent since bilateral

2See Harrison and Rutström (1991).

3

bargaining is assumed.3 Finally, axioms related to changes in the disagreement

points have so far not been considered under multiple-issue bargaining. Here, we

explore the relevance of these disagreement points in bargaining situations under

�xed bargaining trophies. The focus is on the pure contribution of these points

to bargaining solutions where the bargaining sets are not allowed to vary during

the negotiation rounds, in contrast to the traditional Nash bargaining framework.4

More speci�cally, we propose a number of axioms related to issue linkages when the

disagreement points are taken into consideration. This constitutes a main contri-

bution of our model.5

Finally, it is very important to draw the distinction between separate and linked

Pareto e¢ ciency. Classical axioms that are applied to single-issue problems are

based on the idea of separate/local Pareto e¢ ciency, where it is enough for the

solution to be on the Pareto frontier of each set to be e¢ cient6 . In a more general

context, Peters (1985) and Ponsatí and Watson (1997) discuss the idea of global

e¢ ciency in the context of multi-issue bargaining. They argue that e¢ ciency de-

mands that no possible gains from cooperation are lost, which means that each

local solution must belong to Pareto frontier of the sum of the local sets. Given

our context where the issue bargaining set is a simplex, any solution located on the

Pareto frontiers of both sets, say X and Y , maximizes the sum of players�utilities

across issues and is thus also globally Pareto e¢ cient.

3. THE MODEL

Two agents, i = 1; 2, bargain simultaneously over two issues, X and Y . Suc-

cessful bargaining consists in dividing a total payo¤ of 1 for each issue between

the two agents. Failure to achieve an agreement in both issues results in agents

falling back to their disagreement payo¤s in both issues; we denote dXi > 0 (resp.

dYi ) agent i�s payo¤ on issue X (resp. issue Y ). We impose dX1 + dX2 � 1 and

dY1 + dY2 � 1, and denote � = fz 2 R�2+ j0 < z1 + z2 � 1g. The pro�le vector

d = (dX ; dY ) 2 � �� constitutes a linked bargaining problem. We denote by �2

the class of linked bargaining problems.

A linked bargaining solution (or solution), f : �2!R2+�R2+ maps each bargain-ing problem to a payo¤ vector, f(d) = (x; y) � (dX ; dY ) such that x1 + x2 = 1 andy1 + y2 = 1. We interpret

dX2dX1

and dY2dY1to be the agents�relative bargaining powers

3See William Thomson and Terje Lensberg (1989) for single issue models with n-agents.4The envisioned bargaining problems are cases where players know and cannot change the

bargaining set. Of course, this does not preclude negotiations in steps toward the �nal sharing ofthe pre-�xed set. This point is discussed further in the model section.

5Nonetheless, it should be noted that Thomson (1987, 1994) and Youngsub Chun and Thomson(1990, 1992) introduce axioms related to the disagreement point but for single-issue bargainingonly.

6This is the case when both issues are seen separately. The idea of global e¢ ciency only makessense when linkage is considered.

4

over issues X and Y , respectively. For instance, if dX2

dX1is very small (close to zero)

and dY2dY1is large, then player 1 has a strong advantage over issue X but player 2 has

a better bargaining power over issue Y (See Figure 3). Lastly, x1 + y1 and x2 + y2are the overall payo¤s of agent 1 and agent 2, respectively.

[FIGURE 3 HERE]

We introduce two properties that we deem desirable in a solution to a linked

bargaining problem. The �rst axiom stipulates that if the relative bargaining power

is the same across issues, the bargaining rule should respect these relative strengths.

In other words, if both issues "agree" on the relative strengths of the bargainers,

the �nal outcome should respect this overall relative strength.

Axiom 1. "Uniformity" dX2dX1=

dY2dY1

=) x2x1= y2

y1=

dX2dX1.

Next, we require that the agents�total payo¤be independent of how they choose

to allocate their bargaining power across issues.

Axiom 2. "Payo¤ Invariance with respect to bargaining power reallocation across

issues (Invariance)" 8d; d0 2 �2, such that d0X1 + d0Y1 = dX1 + dY1 and d

0X2 + d0Y2 =

dX2 + dY2 ,

x0i + y0i = xi + yi

for i = 1; 2, where (x0; y0) = f(d0).

Note that Invariance can be viewed as having both strategic and normative

content. From a strategic viewpoint, it ensures that agents cannot manipulate

the solution by reallocating their bargaining e¤orts across issues. I.e., in an ex ante

game where agents could revisit their prior investments towards building bargaining

power for each issue, none would �nd an interest to do so. From a normative

standpoint, Invariance ensures that the solution be not partial towards one issue

over the other. Indeed, it asks that disagreement utility play an equivalent role on

each issue, just like agreement utility on each issue has equal weight in each agent�s

total (agreement) payo¤.

These two axioms are not only focused on the role of the disagreement points on

each issue, they also convey the notion of linkage, which is the fundamental distinc-

tion between the linked bargaining problem and the traditional Nash bargaining

problem. We now further illustrate this distinction by showing how linkage would

be ignored if one attempted to treat the linked bargaining problem as a single-issue

bargaining problem. More speci�cally, one may be tempted to combine the two is-

sues as follows: the disagreement utility levels of the players would be D1 = dX1 +dY1

5

and D2 = dX2 + dY2 , respectively, and the size of the cake to be divided would be

2. The reader can easily check that applying, say, the Nash bargaining solution to

this (single-issue) problem yields the following total payo¤s for each agent: xN1 + y

N1

xN2 + yN2

!=

(2 +D1 �D2)=2(2�D1 +D2)=2

!.

The many points in �2 giving rise to the above total payo¤s are of the form:0BBBB@xN1

xN2

yN1

yN2

1CCCCA =

0BBBB@(1 + dX1 � dX2 )=2� c(d)(1� dX1 + dX2 )=2 + c(d)(1 + dY1 � dY2 )=2 + c(d)(1� dY1 + dY2 )=2� c(d)

1CCCCAwith jc(d)j � min

n1�dX1 �d

X2

2 ;1�dY1 �d

Y2

2

ofor all d 2 �2. In particular, taking c � 0

amounts to applying the Nash bargaining solution to each issue independently.

Hence, the Nash bargaining solution entirely ignores the linkage between both is-

sues. In fact, the Nash bargaining solution, whether applied to the joint (single-

issue) problem or to each issue independently, violates Uniformity.7

Taken together, the Uniformity and Invariance axioms characterize a family of

solutions related to what we call the "Linked Disagreement Points solution" (or

LDP solution), which we de�ne as follows:0BBBB@xLDP1

xLDP2

ylDP1

yLDP2

1CCCCA =

0BBBB@D1(1�dX2 )+D2d

X1

D1+D2

D1dX2 +D2(1�dX1 )D1+D2

D1(1�dY2 )+D2dY1

D1+D2

D1dY2 +D2(1�dY1 )D1+D2

1CCCCA ;

where D1 = dX1 +dY1 and D2 = d

X2 +d

Y2 . This solution takes its name from the fact

that it "links" the disagreement vectors of each issue. This can be seen graphically

in Figure 4.

[FIGURE 4 HERE]

7The reader can easily check that xN2xN1

=yN2yN1

=dX2dX1

=dY2dY1

only when dX2dX1

=dY2dY1

= 1.

6

Theorem 1. A solution satis�es Uniformity and Invariance if and only if it is

a payo¤-equivalent variant of the LDP solution:0BBBB@x1

x2

y1

y2

1CCCCA =

0BBBB@xLDP1 � c(d)xLDP2 + c(d)

ylDP1 + c(d)

yLDP2 � c(d)

1CCCCA (1)

with c : �2 ! R such that c(d) = 0 whenever dX2dX1=

dY2dY1.

Proof. The reader can check that such a solution satis�es Uniformity and In-

variance. Conversely, consider a solution satisfying both axioms. By Invariance, the

total payo¤ of each agent only depends on each agent�s overall bargaining power,

Di = dXi + d

Yi . Now consider an alternative pro�le, d

0, such that d0X2d0X1

=d0Y2d0Y1

with

d0X1 + d0Y1 = dX1 + dY1 and d

0X2 + d0Y2 = dX2 + d

Y2 (See Figure 5).

[FIGURE 5 HERE]

Note that d0X2d0X1

=d0Y2d0Y1

=d0X2 +d0Y2d0X1 +d0Y1

. By Uniformity, x01 = y01 =

d0X1 +d0Y1d0X1 +d0Y1 +d0X2 +d0Y2

=

D1

D1+D2and x02 = y02 =

d0X2 +d0Y2d0X1 +d0Y1 +d0X2 +d0Y2

= D2

D1+D2. Invariance yields x1 + y1 =

x01+y01 =

2D1

D1+D2= xLDP1 +yLDP1 and x2+y2 = x02+y

02 =

2D2

D1+D2= xLDP2 +yLDP2 .

Thus, the solution can be written as in Expression (1) with Uniformity ensuring

that c(d) = 0 whenever dX2dX1=

dY2dY1.

4. RELAXING THE AXIOMS

We now present what solutions are permitted when dropping the Uniformity

and Invariance axioms.

4.1. Dropping Uniformity

The role of the Uniformity axiom in the proof of Theorem 1 was to pin down the

total payo¤ that the solution must assign to each agent. Hence, requiring Invariance

alone characterizes a class of solutions assigning a total payo¤ that only depends

on each agent�s overall bargaining power.

Theorem 2. A solution satis�es Invariance if and only if it can be written as

follows: x1 + y1

x2 + y2

!=

g1(D1; D2)

g2(D1; D2)

!,

where Di = dXi + dYi , i = 1; 2.

Proof. This follows directly from the Invariance axiom.

7

Many solutions satisfy Invariance, including the well-known Nash bargaining

solution. However, because the Invariance axiom is only concerned with aggregate

bargaining power, it provides little indication on how to link the bargaining issues,

X and Y . Quite to the contrary, Invariance dictates to what extent the issues can

be treated as a single one. Hence, the Uniformity axiom is a crucial one to explore

issue linkage. In what follows, we replace the Invariance axiom by weaker ones and

explore the type of solutions a¤orded by the Uniformity axiom.

4.2. Dropping Invariance

The Invariance axiom has strong implications for the nature of the solution.

While Invariance is consistent with the additive framework under study, one may

wish to explore the possibilities that dropping the Invariance axiom a¤ords. Clearly,

the Uniformity axiom alone allows for too many solutions to be of interest, so we

shall combine it with other mild axioms.

Keeping with the spirit of impartiality, we argue that a solution should not

behave di¤erently across issues. More precisely once bargaining power has been

taken into account, via the agents� issue-wise disagreement points, the solution

treats both agents and issues symmetrically.

Axiom 3. "Issue neutrality" y1�dY1x1�dX1

=y2�dY2x2�dX2

This axiom is an axiom of neutrality vis-a-vis the issues. For example, ify1�dY1x1�dX1

>y2�dY2x2�dX2

, the solution confers an a priori advantage to player 1 over player

2 in issue Y , which can be viewed as undesirable. Therefore, this condition must

hold at equality to ensure neutrality with respect to issues once bargaining powers

are accounted for.

Next, we ask that a solution be consistent: achieving an agreement in several

steps rather than in a single round should not a¤ect the outcome. This axiom

requires de�ning an intermediate linked bargaining problem, where the payo¤ to

be divided in issue X (resp. Y ) is EX � 1 (resp. EY � 1). A triple (d;EX ; EY ) 2�2� (0; 1]2 is an intermediate linked bargaining problem (or intermediate problem)

if dX1 + dX2 � EX and dY1 + d

Y2 � EY . The domain of a solution is naturally

extended to account for intermediate problems.

Axiom 4. "Composition" f(d) = f(f(d;EX ; EY )) for any intermediate prob-

lem (d;EX ; EY ).

The next requirement is one of smoothness, which ensures that the solution be

not wildly sensitive to changes in the bargaining powers:

Axiom 5. "Smoothness" f is continuously di¤erentiable in d.

Requiring Axioms 3-5 in addition to Uniformity yields a family of bargaining

solutions:

8

Theorem 3. A solution satis�es Uniformity and Axioms 3�5 if and only if:

�2 ! R+ [ f+1gd 7! x2�dX2

x1�dX1

,

is a continuously di¤erentiable function such that:

i) x2�dX2x1�dX1

=dX2dX1

if dX2

dX1=

dY2dY1, and

ii) x02�d0X2

x01�d0X1=

y02�d0Y2

y01�d0Y1=

x2�dX2x1�dX1

for all (d0X ; d0Y ) 2 (dX ; x)�(dY ; y),8 where (x0; y0) =f(d0).

Proof: The reader can readily check su¢ ciency, but the proof of necessity, pro-viso ii), requires several steps. Let f be a bargaining solution satisfying Uniformity

and Axioms 3 through 5. Let d 2 �2 and denote (x; y) = f(d).Claim 1: For all d0 = (d0X ; d0Y ) 2 [dX ; x]� [dY ; y],9 the following holds:

(a) f(d0X ; dY ) = (x; y);

(b) f(dX ; d0Y ) = (x; y); and,

(c) f(d0) = (x; y):

Let d 2 �2 and let (d0X ; d0Y ) 2 [dX ; x] � [dY ; y]. We �rst prove point (a). ByComposition, y = fY (f(d; d0X1 + d0X2 ; 1)) = f

Y (d; d0X1 + d0X2 ; 1) because the coordi-

nates of the latter term already sum up to 1. By Issue Neutrality,����������������!dXfX(d; d0X1 + d0X2 ; 1)

is colinear to����������������!dY fY (d; d0X1 + d0X2 ; 1) which, together with the fact that f

Y (d; d0X1 +

d0X2 ; 1) = y and the fact that��!dXx and

��!dY y are colinear, implies that

����������������!dXfX(d; d0X1 + d0X2 ; 1)

and��!dXx are colinear. Lastly, the fact that the coordinates of fX(d; d0X1 + d0X2 ; 1)

sum up to d0X1 + d0X2 implies that fX(d; d0X1 + d0X2 ; 1) = d0X . Finally, by the Com-

position axiom, fX(f(d; d0X1 + d0X2 ; 1)) = x, yielding the result.

Note that, by assumption on f , x � d0X and y � d0Y . It follows that the

rays (dX ; x) and (dY ; y) are positively sloped, implying x2�dX2x1�dX1

2 R+ [ f+1g. By

Smoothness, (dX ; dY ) 7! x2�dX2x1�dX1

is continuously di¤erentiable.

An analogous argument leads to f(dX ; d0Y ) = (x; y). Finally applying (a) to

the latter expression leads to f(d0X ; d0Y ) = f(dX ; d0Y ) = (x; y), proving point (c).

Claim 2 For all d0X 2 (dX ; x)\� and all d0Y 2 (dY ; y)\�, the following holds:

(a) f(d0X ; dY ) = (x; y);

(b) f(dX ; d0Y ) = (x; y); and

(c) f(d0) = (x; y).8 (dY ; y) denotes the line passing through dY and y.9 [dY ; y] denotes the line segment connecting dY to y.

9

We �rst prove statement (a). Let d 2 �2. The line (dX ; x) divides � into two

convex regions, �+ and �� such that �+ \�� = (dX ; x) \�. (See Figure 6)

[FIGURE 6 HERE]

Let d0X 2 (dX ; x) \ � and suppose d0X =2 [dX ; x] (the case not covered by

Claim 1). We shall show that f(�; dY ) is stable on each of the subsets �+ and

��. Indeed, suppose there existed d̂X 2 ��n�+ such that f(d̂X ; dY ) 2 �+n��.For any � 2 [0; 1] denote d�;X = �dX + (1 � �)d̂X : By Continuity of f in d,lim�!1 f

X(d�;X ; dY ) = x 2 ��. Yet, by Composition, it must be that [d�;X ; fX(d�;X ; dY )]\[dX ; x] = ; for any � < 1. Otherwise, there would exist some �dX 2 [d�;X ; fX(d�;X ; dY )]\[dX ; x], for which Claim 1 would imply f( �dX ; dY ) = x and, by Composition, we

would have f(d̂X ; dY ) = (f( �d; dY ) = x, contradicting the fact that f(d̂X ; dY ) 2�+n��. Finally, because [d�;X ; fX(d�;X ; dY )] \ [dX ; x] = ; for any � < 1, the

convexity of �� implies that ClffX(d�;X ; dY )j0 � � < 1g \ fxg = ;; where Cl isthe closure operator, implying that lim�!1 f

X(d�;X ; dY ) 6= x, a contradiction.Statement (b) is proved in a similar fashion as statement (a), and (c) is obtained

by combining (a) and (b), as was done for Claim 1�Theorem 3 provides the general structure of linked bargaining solutions satisfy-

ing Uniformity and axioms 3 through 5. In addition, one may �nd it desirable that

the improvement of an agent�s bargaining power in either issue should not hurt her

overall payo¤:

Axiom 6. "Monotonicity" For all d; d0 2 �2,(d0i � did0j = dj

=) x0i + y0i � xi + yi

where (x0; y0) = f(d0):

In order to state the next Theorem, we de�ne the function a : d 7! x2�dX2x1�dX1

on

�2. a(d) can be interpreted as the ratio of relative gains of agent 2 over agent 1 on

issue X (and, therefore, on issue Y as well).

Corollary 1. A solution satis�es Uniformity and Axioms 3�6 if and only if:

@a@dX1

� x2�dX2(x1�dX1 )A

@a@dX2

� � 1A

@a@dY1

� x2�dX2(x1�dX1 )A

@a@dY2

� � 1A

10

where A = x1 + y1 � dX1 � dY1 , in addition to the conditions of Theorem 3.

Proof : We show the �rst inequality. Let f satisfy axioms 1-5. Let d 2 �2,and " > 0 such that (dX1 + "; d

X2 ; d

Y1 ; d

Y2 ) 2 �2. Denote � = a(dX ; dY ), (x0; y0) =

f(dX1 +"; dX2 ; d

Y1 ; d

Y2 ) and �

0 = a(dX1 +"; dX2 ; d

Y1 ; d

Y2 ). By de�nition of a(�), x02�dX2 =

�0(x01 � dX1 � ") and x0Y2 � dY2 = �0(fY1 � dY1 ). Adding both equalities yields

x02 + y02 � dX2 � dY2 = �0(x01 + y

01 � dX1 � dY1 � "). The same operation applied to

the original bargaining problem yields x2 + y2 � dX2 � dY2 = �(x1 + y1 � dX1 � dY1 ).Subtracting the latter equality from the previous one yields x02 + y

02 � x2 � y2 =

�(x1+y1�dX1 �dY1 )��0(x01+y01�dX1 �dY1 �"). Using the fact that x01+y01+x02+y02 =x1 + y1 + x2 + y2 = 2 leads to:

x01 + y01 � x1 � y1 = �(x1 + y1 � x01 � y01) + (�� �0)(x01 + y01)

�(�� �0)(dX1 + dY1 ) + �0"

(1 + �0)x01 + y

01 � x1 � y1"

= �0 +(�� �0)

"(x1 + y1 � dX1 � dY1 ):

Taking the limit towards " = 0 leads to:

(1 + �)@(x1 + y1)

@dX1= �� @a

@dX1� (x1 + y1 � dX1 � dY1 ):

It follows from this last expression that imposing monotonicity (@(x01+y

01)

@dX1� 0)

amounts to requiring � � @a@dX1

(x1 + y1 � dX1 � dY1 ) � 0, as was to be proven. Theother inequalities are proven similarly.�

Several solutions stand out among the ones satisfying Uniformity and Axioms

3-6. For instance, any rule taking a convex combination of the relative bargaining

powers in each issue, such that a(dX ; dY ) = �dX2

dX1+ (1 � �)d

Y2

dY1for some � 2 [0; 1],

belongs to this class. We call this the class of monotonic equal net ratio solutions

(See Figure 7 ).

[FIGURE 7 HERE]

This class consists of a continuum of solutions of which an extreme case stands

out. The single-issue dictatorship solution requires bargaining gains be allocated

according to the relative bargaining powers over issue X (i.e.,dX2

dX1) only. In other

words, the bargaining power dY2dY1over issue Y does not matter (See Figure 8).

[FIGURE 8 HERE]

11

The LDP solution could be seen as a re�nement, where the gains on each issue

depend on the absolute bargaining powers of each agents: a(dX ; dY ) = dX2 +dY2

dX1 +dY1,

which amounts to de�ning the convex combination as �(dX ; dY ) = dX1dX1 +d

Y1. Graph-

ically and as was discussed earlier, this solution links both disagreement points dX

and dY , and locates the solution outcome on the Pareto frontier of each bargaining

set (see Figure 2). Thus the LDP solution could be seen a balanced compromise so-

lution since it combines the bargaining powers over both issues: it takes the global

bargaining power ratio between both players to determine the outcome.

It is noteworthy that the degrees of freedom granted by the class of monotonic

equal net ratio solutions is "horizontal", in the sense that linkage is not a question

of how strongly the two issues are linked, but a question of how much weight is given

to the relative bargaining powers in each issue. In particular, a solution treating

both issues separately would not belong to the class. This can be seen with the

(single issue) Nash bargaining solution, for instance, which would correspond to

a � 1 at all pro�les, thus violating Uniformity as was demonstrated ealier. In otherwords, "no linkage" is not a special case of linkage.

5. CONCLUDING REMARKS

Stylized facts suggest that in international law, issues pertaining to commerce

and environment are usually dealt with in a con�icting manner. This has been

a trend since 1972 when the United Nations Environment Program (UNEP) was

established. That year was the year of the United Nations� conference on the

environment held in Stockholm, and is now seen as a turning point in international

environmental awareness. The con�icting nature of international environmental law

stems from the fact that trade and environmental concerns carry trade-o¤s. The

GATT (WTO after 1995) is in general against unilateral discriminatory measures,

as per Article XX. However, if these measures are required by an international

environmental agreement (IEA) then the issue becomes more problematic because

simultaneous negotiations are needed. Indeed, the class of monotonic equal net

ratio solutions, which takes a convex combination of the relative bargaining powers

in each issue, seems to re�ect the way simultaneous bilateral bargaining over trade

and environment has been taking place. In this example, if environmental measures

are not in con�ict with WTO�s Article XX then a solution in the spirit of the

single-issue dictatorship solution requires bargaining gains be allocated according

to the relative bargaining powers over the trade issue only (See, e.g. the 1991

GATT tuna case pitting Mexico versus USA, and the 2001 WTO Shrimp case

pitting the USA versus Malaysia, Philippines, Pakistan and India). In this case

there is precedence of the older treaty, that is the GATT/WTO. Otherwise, a

12

convex combination of relative powers over both issues will determine the �nal

outcome as was the case with the Genetically Modi�ed Organisms (GMOs) con�ict

in 2003 between the USA, Canada, Argentina on one hand and the EU on the

other.10 During this con�ict, an IEA� the Cartagena protocol on bio-safety� was

used to challenge WTO rules; in other words, a convex combination of trade and

environment negotiation powers shaped the �nal solution of the con�ict. In this case

there is precedence for the more precise treaty, that is the Cartagena protocol. Yet,

this precedence is not absolute because the older treaty, which is on trade, still has

jurisdiction. Moreover, the monotonic equal net ratio solutions may also inform

us about the future resolution outcome of the aviation emissions dispute pitting

EU versus non-EU countries. Because aviation emissions were recently included

into the European Emission Trading Scheme (ETS), non-EU airlines operating

international routes will also have to comply with the ETS. In response, non-EU

countries are considering retaliatory measures invoking trade sanctions and calling

upon the European Court of Justice for a ruling. As in the Cartagena dispute a

monotonic equal net ratio solution can be expected given its desirable properties

for the bargaining countries.11

10http://www.wto.org/english/tratop_e/dispu_e/cases_e/ds291_e.htm11For more information about the aviation emissions case in the EU see the July 2011 Newsletter

of the International Center for Climate Governance.

13

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15

FIGURES

Figure 1. The feasible set X when normalized to unity

Figure 2. The Linked Disagreement Points Solution

16

Figure 3. The two-issue bargaining problem

Figure 4. The LDP and Nash solutions

17

Figure 5.

Figure 6.

18

Figure 7. The monotonic equal net ratio solution when � = 1=2:

Figure 8. Single-issue dictatorship (issue X)

19